Limited cognitive ability and selective information processing

Games and Economic Behavior 120 (2020) 345–369

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Games and Economic Behavior www.elsevier.com/locate/geb

Limited cognitive ability and selective information processing ✩ Benson Tsz Kin Leung Faculty of Economics, University of Cambridge, UK

a r t i c l e

i n f o

Article history: Received 6 December 2018 Available online 4 February 2020 JEL classification: D83 D90 Keywords: Limited ability Information overload Information avoidance Confirmation bias Wishful thinking Polarization

a b s t r a c t This paper studies the information processing behavior of a decision maker (DM) who can only process a subset of all information he receives: before taking an action, the DM receives sequentially a number of signals and decides whether to process or ignore each of them as it is received. The model generates an information processing behavior consistent with that documented in the psychological literature: first, the DM chooses to process signals that are strong; second, his processing strategy exhibits confirmation bias if he has a strong prior belief; third, he tends to process signals that suggest favorable outcomes (wishful thinking). As an application I analyze how the Internet and the induced change in information availability affects the processing behavior of the DM. I show that providing more/better information to the DM could strengthen his confirming bias. © 2020 Elsevier Inc. All rights reserved.

1. Introduction There is abundant evidence that people selectively process information in a systematically biased way. For instance, investors avoid looking at their financial portfolios when the market is down; individuals tend to ignore information that challenges their existing beliefs; people tend to attend to information that support desired outcomes but ignore contradictory evidences (see Golman et al. (2017) for a review of the literature of information avoidance). These behaviors of selective information avoidance lead to biased decision making and could significantly worsen our well-being as we miss out useful information. As bad as information avoidance sounds, we are not able to process all available information, especially in this information era. Unavoidably, everyday we have to make many decisions on whether to process or ignore pieces of information. Processing a piece of information allows us to better understand its content and update our beliefs but consumes cognitive resources like time and attention which are limited. This limitation imposes a constraint on our processing capacity,1 which gives rise to the possibility of information avoidance: it could be optimal for individuals to ignore a piece of information in order to save their cognitive resources for another piece of information.

✩ I would like to thank Professor Jacques Crémer for helpful comments throughout the project. Part of the project was done during my stay in the Behavior and Inequality Research Institute in Bonn. I would like to thank the administrative staffs, students and professors for their kind hospitality. I also thank Olivier Compte, Lorenz Götte, Renato Gomes, Johannes Hörner, Jean Tirole, two anonymous referees and the audiences in various seminars and conferences for helpful comments and suggestions. The financial support from the Digital Chair Jean-Jacques Laffont is gratefully acknowledged. E-mail address: [email protected]. 1 For instance, during an election, there are millions of articles on the Internet that could guide our voting decision. However, we are endowed with limited time and attention such that we could only read a very small subset of all the available articles.

https://doi.org/10.1016/j.geb.2020.01.005 0899-8256/© 2020 Elsevier Inc. All rights reserved.

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It is increasingly important nowadays to study individuals’ selective processing behavior as information overload becomes a prominent issue. There is no doubt that the advance in technology provides us with more information on different issues. However, the amount of available information clearly exceeds our processing capacity and we have to strategically use our scarce cognitive resources. Indeed, there is evidence that the Internet are associated with biased processing behavior (Flaxman et al. (2016)). In order to understand the impacts of the Internet, information policies, or in general changes in informational environment, it is important to understand how people process information when they have limited processing ability, and how their processing behavior changes with the policies. To answer these questions, this paper proposes a simple model of sequential information processing. Consider a decision maker (DM) who wants to match his action to an unknown state of the world, e.g., vote for a candidate if he/she is the best option, invest in a project if it is profitable, etc. Before taking the action, the DM receives sequentially T imperfect signals about the state of the world. However, he is endowed with limited cognitive resources such that he can only “process” T¯ (< T ) signals. In each period t = 1, · · · , T , the DM observes some preliminary (imperfect) information about the realization of a signal, e.g., the title of an article, and decides whether to “process” it. If he processes, he learns perfectly its realization and updates his belief; otherwise, he “ignores” the signal without updating his belief. In other words, in each period, he decides whether to update his belief with the signal based on his (imperfect) knowledge about its realization.2 Once he has consumed all his cognitive resources or has received all T pieces of information, he takes an action. The model resembles a game of sequential search of information with a constraint on processing capacity. Given the capacity constraint, there is an trade off between spending a unit of capacity on the current signal or some future signal. That is, there is a loss of processing as DM could save the unit of capacity for some future signal; while there is a gain of processing as otherwise the DM will not take into account the current signal in decision making and may take suboptimal action. The loss and gain of processing varies with the realization of the current signal, which drives selective processing decisions. I show that in the equilibrium, the DM selectively ignores some of the signals he receives. Moreover, the equilibrium processing strategy is “asymmetric”, i.e., the processing decisions are in general different for belief-confirming and belief-challenging signals. In the continuation game where the capacity is equal to 1, i.e., T¯ = 1, the game becomes a simple stopping time problem. The equilibrium processing strategy of the DM exhibits behavioral phenomena that are well-documented in the empirical and experimental literature. First, it exhibits a preference for strong signals (Itti and Baldi (2006)), i.e., the DM tends to process strong signals and ignore weak signals. Second, the equilibrium processing strategy of confident individuals exhibits confirmation bias (Kahan et al. (2012)), i.e., if the DM a priori strongly believes that one state is more probable than the other, he tends to process information which confirms his existing belief and ignore belief-challenging information. Third, the equilibrium processing strategy exhibits wishful thinking (Krizan and Windschitl (2007)), i.e., if one state is much more desirable than the other as it is associated with a much higher maximum payoff, the DM tends to process information which supports the more desirable state. The results suggest that these biases could be understood as optimal strategies, driven by the limitation in processing ability. Lastly, I analyze how changes in information structures, for example induced by the Internet or information policies, affect the processing behavior of individuals. I show that providing more or in average better information to the DM could strengthen his confirmation bias, i.e., he has more incentive to process belief-confirming information and ignore beliefchallenging information. Even if two individuals are exposed to the same sequence of signals, they have more incentive to “cherry pick” the information that confirms their existing belief which could promote polarization. Moreover, media has more incentive to selectively publish biased news stories as the demand increases. These results explain a number of empirical phenomena documented in the literature of ideological polarization. The rest of the paper is organized as follows. In the next section, I present a review of the related literature. Section 3 shows a simple version of the model to illustrate the assumption of bounded rationality and intuitions of the results. In section 4, I present the model setting. Section 5 shows the results where T¯ > 1 while section 6 turns to the continuation game where T¯ = 1. Section 7 shows two simple applications. Lastly, I conclude. 2. Related literature This paper is related to a wide range of literature, spanning economics, political science and psychology. First of all, the core assumption of this paper, i.e., the DM has to use his limited cognitive resources in order to update his belief, is built on psychological theories and evidence. More specifically, it could arise from different channels as suggested by the literature, including the effort required to understand and to memorize the information.3

2 This is in particular different from models of information acquisition and rational inattention, where the DM decides whether to incur a fixed cost to acquire a piece of information before knowing its realization. By definition, the DM’s acquisition decisions cannot be different for belief-confirming and belief-challenging information. 3 Memorizing a piece of information involves an encoding process, which can be strengthen by different factors, including time (Goldstein et al. (2011)), attention (Shallice et al. (1994), Craik et al. (1996), Benjamin and Bjork (2000) and Uncapher and Rugg (2005)), and how “deep” the individual processes the information (Craik and Lockhart (1972), Craik and Tulving (1975), Wagner et al. (1998) and Brewer et al. (1998)). Strengthening the encoding process increases the probability of recalling the information, but consumes scarce cognitive resources.

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On one hand, cognitive psychological theories proposed by Langdon and Coltheart (2000), Coltheart et al. (2011) and Connors and Halligan (2015) suggest that beliefs are formed as explanations to information. It requires efforts to understand the information in order to integrate it with the individual’s existing system of beliefs. On the other hand, there is evidence from psychological studies showing that memory of information plays a big role in belief formation. One example is the seminal study of availability heuristic by Tversky and Kahneman (1973). They find that individuals evaluate the probability of an event by how much and how easily supportive evidence can be retrieved from memory. Memory-based belief formation is also modeled in the theoretical economics literature to explain different behavioral phenomena, e.g., Bénabou and Tirole (2002) and Baliga and Ely (2011). Both mechanisms, i.e., to understand or to memorize information, imply that individuals have to spend cognitive resources to “process” the information in order to update their belief. The idea of limited cognitive ability has been introduced in the fast-growing literature of rational inattention, e.g., Sims (2003), Matejka and McKay (2014) and many others. They study a static4 problem of information acquisition where the cost is proportional to the reduction in Shannon entropy. Different from this paper, under the setup of information acquisition, they cannot account for the phenomena that individuals selectively update their beliefs based on the realization of the signals, and that different individuals with different prior beliefs may update differently when they receive the same signals. Moreover, as the cost depends only on individuals’ choices of the distribution of their posterior beliefs, the role of information structure is limited, if not entirely absent. As a result, the literature of rational inattention is not suited to shed light on how changes in informational environment affects the belief formation of individuals.5 In terms of results, this paper contributes to the literature of information avoidance (see Golman et al. (2017) for an extensive review of the literature). For instance, Eil and Rao (2011) show that individuals update less when they receive negative information about their appearance or intelligence than when they receive positive information. Karlsson et al. (2009) and Sicherman et al. (2016) find that investors avoid checking their financial portfolios when the market is down. The papers mentioned above, alongside with many others, document different systematic biases in the processing behavior of individuals, including the well-known confirmation bias and wishful thinking. There are many economic theories that explain the confirmation bias in processing behavior. For instance, Akerlof and ˝ Dickens (1982), Koszegi (2003), Brunnermeier and Parker (2005) and Caplin and Leahy (2019) show that anticipatory utility or belief-dependent utility leads to the confirmation bias; Carrillo and Mariotti (2000) and Bénabou and Tirole (2002) show that confirmation bias can be used as a remedy for time inconsistent preferences; Crémer (1995) and Aghion and Tirole (1997) explain it with interpersonal strategic concerns; and the list goes on. On the other hand, Heller and Winter (2019) show that wishful thinking can serve as a commitment device in strategic environment. In contrast, this paper belongs to a relatively small, but growing, set of literature which suggests limitation in cognitive ability explains a number of behavioral biases in belief updating (Compte and Postlewaite (2012), Wilson (2014) and Jehiel and Steiner (2018)). Compared to the existing literature, this paper derives a wider range of results. Not only it explains the existence of biases as an optimal strategy, it also shows how biased processing behavior changes with individual characteristics and the informational environment, which sheds lights on the heterogeneity of biases among individuals and the effects of technological advancement or information policies. Lastly, the results in this paper shed light on different issues in the information era. It includes political polarization, which receives lots of attention in recent years (see Prior (2013) for a review). On one hand, Gentzkow and Shapiro (2011) and Flaxman et al. (2016) show that online media expose individuals to belief-challenging information. On the other hand, there is evidence that the political ideology among US citizens is getting polarized (Bartels (2000), Flaxman et al. (2016)), especially among those who are more politically engaged and partisan (Baldassarri and Gelman (2008), Abramowitz and Saunders (2008), Hetherington (2009).). This paper shows that the limitation in processing ability, or information overload, could explain the aforementioned wide range of phenomena. 3. An illustrative example Consider a voter who must decide to vote for either a left wing or a right wing candidate. Only one of the candidates is “good”. Voting for the “good” candidate yields one util while voting for the “bad” candidate yields zero util. Before he receives any information, the voter believes that there is a 70% probability that the left wing candidate is the good candidate. Before he must vote, he knows that he will receive two tweets from two journalists whom he trusts, in which each provides information about the quality of the candidates. More precisely, the tweets could be left leaning, right leaning or neutral. When the left wing candidate is the good candidate, the tweets are more likely to be left leaning than to be neutral, and are more likely to be neutral than to be right leaning. Similarly, when the right wing candidate is the good candidate, the tweets are more likely to be right leaning than to be neutral, and more likely to be neutral than to be left leaning. (The two tweets are independent conditional on the identity of the good candidate.) The information structure is represented in Table 1.

4 The literature of rational inattention gets around the mechanism of belief formation and assume the cost of reduction in uncertainty is proportional to the reduction in entropy. As a result, receiving several weak supportive evidence is equivalent to receiving one strong supportive evidence, which overlooks the dynamics of information processing and belief formation. In contrast, this paper focus on the dynamics of information processing. 5 Under the setting of rational inattention, one obvious but restricted way to model a change in informational environment is to assume an arbitrary change of the cost function, which is difficult to interpret and justify.

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Table 1 The probabilities of the tweets given the identity of the good candidate. The tweet left leaning

neutral

right leaning

left wing candidate is good

0.5

0.3

0 .2

right wing candidate is good

0. 2

0.3

0 .5

Table 2 The voter’s belief and voting decision given the tweet he had processed. The tweet he had processed left learning

neutral

/ ≈ 85%

right leaning

/ = 70%

/ ≈ 48%

the voter’s belief that the left wing candidate is good

35 41

7 10

14 29

voting decision

left wing

left wing

right wing

Now, the voter knows that he can “process” and use only one tweet for his voting decision. After he sees the first tweet, he knows whether it is left leaning, neutral or right leaning, but has to read the attached long article and understand its argument if he wants to update his belief and use the tweet for his voting decision. “Processing” the tweet takes time, so if the voter processes the first tweet, he will not be able to process the second tweet and will choose whom he votes for based on the first tweet. On the other hand, if he ignores the first tweet, he processes the second tweet and decides which candidate he will vote for based on it. To summarize, the voter faces a capacity constraint on belief updating and trade-off between processing the first or the second tweet. Beliefs and voting decisions For simplicity, I assume the voter updates his belief with the first or the second tweet in the same way, using the following Bayesian formula:

⎛

Belief given the processed   left leaning = neutral tweet is right leaning





⎞

left leaning ⎜ ⎟ neutral a tweet is ⎜ ⎟ Pr ⎝ ⎠ right leaning and the left wing candidate is good

⎛



Pr ⎝a tweet is

left leaning neutral right leaning

⎞ ⎠

It implies that the voter updates his belief with the processed tweet as if he has only received that tweet. If the voter has ignored the first tweet, he does not infer information about the tweet from his processing decision and updates his belief as if he has only received the second tweet. The online appendix relaxes this assumption and shows that the results do not change qualitatively.6 Given this assumption, the voter’s belief and his voting decision given the tweet he had processed is presented in Table 2. Processing decision Now I analyze the voter’s information processing decision. First consider the case where the first tweet is left leaning. If the DM processes the tweet, he will not be able to process the second tweet and will vote for the left wing candidate. The expected utility of processing the left leaning tweet is therefore equal to the conditional probability that the left wing candidate is good given the fact that the tweet is left leaning:



Pr



left wing  first tweet  candidate  is left  is good leaning



=

35 41

.

On the other hand, if the voter ignores the left leaning tweet, his voting decision will depend on the second tweet, which is summarized in Table 2. The expected utility of ignoring the left leaning tweet is therefore equal to a weighted average of the expected utility of his voting decisions given different realizations of the second tweet:

⎛

⎞

⎛

⎞

second second  left wing  first tweet tweet is ⎟  ⎜ tweet is ⎟ ⎜ Pr ⎝ ⎠ × Pr ⎝ candidate  is left & ⎠ left left  is good leaning leaning leaning 6 In this illustrative example, the result holds as long as the voting decision of the voter given the tweet he had processed follows Table 2, regardless of whether he has processed the first or the second tweet.

B.T.K. Leung / Games and Economic Behavior 120 (2020) 345–369





second +Pr tweet is neutral ⎛ second ⎜ tweet is +Pr ⎝ right leaning





349



left wing  first tweet second  × Pr candidate  is left & tweet is  is good leaning neutral ⎞ ⎛ ⎞ second  right wing  first tweet tweet is ⎟  ⎟ ⎜ ⎠ × Pr ⎝ candidate  is left & ⎠ right  is good leaning leaning

Note that when the voter’s first period self evaluates the voting decisions of his second period self, he takes into account his knowledge of the first tweet; while his second period self will take into account only the second tweet. Therefore, there is an inconsistency between his first and second period self in the way they evaluate the two candidates and voting decisions. With some simple algebra, the expected utility of ignoring the first leaning tweet is equal to 31/41, which is smaller than the expected utility of processing. The voter processes the left leaning tweet. With similar computations, the expected utilities of processing and ignoring a neutral tweet are equal to 7/10 and 71/100 respectively; while the expected utilities of processing and ignoring a right leaning tweet are equal to 15/29 and 187/290 respectively. Therefore, the voter ignores the first tweet if it is neutral or right leaning. To summarize, the voter processes the first tweet if and only if it is left leaning, i.e., it confirms his belief. The processing decisions of the voter exhibit a confirmation bias, which is well documented in the literature of information avoidance. Most of the theories in the literature explain the bias by belief-dependent utility, exogenous biases or interpersonal interaction. In contrast, the voter in this example is rational, Bayesian and cares only about maximizing the probability of voting for the good candidate. In this example, the confirmation bias is an optimal strategy solely driven by the limitation in processing ability. Intuition When the voter makes his processing decisions, he trades off between the loss and gain of processing the tweet. On one hand, there is a loss of processing because there is an opportunity cost of forgoing the second tweet; on the other hand, there is a gain of processing because otherwise the voter will not take into account the first tweet and may make suboptimal voting decisions. The loss and gain of processing the tweet varies with its realizations, which induce selective processing decision. When the voter receives a left leaning first tweet, he becomes more confident about the voting for the left wing candidate, which reduces the importance of the second tweet. In fact, in this special example, the second tweet becomes useless as it is not strong enough to alter the optimal voting decision. Even if the second tweet is right leaning, the optimal decision is still to vote for the left wing candidate. On the other hand, there is a gain of processing as otherwise the voter will become under-confident about the left wing candidate and will switch sub-optimally to voting for the right wing if the second tweet is right leaning. When the voter receives a neutral tweet, the loss and gain are different. In particular, there is no gain of processing as the voter still have the correct belief after ignoring the tweet. On the other hand, the second tweet is in average informative and processing it will (in average) improve the voting decisions of the voter. Lastly, when the voter receives a right leaning tweet, the voter believes that the right wing candidate is marginally better. He is not confident about which action is optimal. In this case, the second tweet becomes more important and the loss of processing is big, and it outweighs the gain of processing. In general, there is an asymmetry in the processing behavior regarding belief-confirming and belief-challenging information, because of the differences in loss and gain as illustrated above. However, the asymmetry does not always lead to confirmation bias. As will be shown in section 6, in the continuation game of the general model where T¯ = 1, there is a confirmation bias when the voter is a priori confident about the state, but it is otherwise not necessarily true. 4. Model setting Primitives The decision maker faces a binary choice problem with two actions, a ∈ {l, r }. For example, l and r could represent voting for the left wing and right wing candidate respectively. There are two possible states of the world, ω ∈ { L , R }. The DM wants to match his action to the state, i.e., his utility function u (a | ω)7 follows

7

The model is identical up to the following transformation in the utility function:

u (l | L ) = u L + A ;

u (l | R ) = B ;

u (r | L ) = A ;

u (r | R ) = u R + B ,

for any constants A , B ∈ R . What matters are the difference between the utilities of the two actions, fixing the state. The model can thus be applied to a setting where action l is risky, i.e., it gives positive payoff u L in state L and negative payoff −u R in state R, while action r is safe and gives 0 payoff in both states.

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u (l | L ) = u L > 0; u (r | R ) = u R > 0; u (l | R ) = u (r | L ) = 0. The DM’s prior belief is ( p L , p R ), with p L + p R = 1 and p L , p R > 0. Without loss of generality, I assume u L p L ≥ u R p R , i.e., action l is a priori optimal. Before the DM takes the action, he looks for information about the state of the world, for example on Google, Facebook or traditional media. The available information, for example articles on the Google news page, is represented by a finite8 sequence of signals, denoted (st )tT=1 . A signal st is drawn with p.d.f. f t L and f t R in states L and R respectively. Without loss of generality, a signal is identified with its likelihood ratio9 :

st :=

f t L (st ) f t R (st )

.

There are no signals that perfectly reveal the state such that st ∈ / {0, ∞}. I allow for the possibility that the p.d.f. f t L , f t R vary with t, i.e., different signals along the sequence may be drawn from different information structures. For example, articles on different positions of the Google news page could have different qualities, as they are ranked based on their features by an algorithm. For technical convenience, I assume that the c.d.f. F t L and F t R are continuous and f t ω spans R++ :

f t ω (st ) > 0 for all st ∈ (0, ∞) and both ω = L , R , such that there are arbitrarily strong signals supporting the two states, but most results hold qualitatively for more general distributions.10 I call a signal st > 1 a belief-confirming signal as it supports state L and confirms the DM’s prior belief that action l is optimal; while a signal st < 1 supports state R and is belief-challenging.11 A signal st = 1 is pure noise which provides no information about the state of the world. Strength of signals The strength of a signal st , denoted by STR(st ), is defined as follows:



STR(st ) = max st , st−1 ∈ [1, ∞) . The larger st , the more likely that the signal is drawn in state L instead of state R, and the more convincingly it supports state L; similarly, the larger st−1 , the more convincingly it supports state R. Analogously, the strength of a set of signals E ⊂ (0, ∞) is defined as:

   f t L (st )dst f t R (st )dst E E STR(st ∈ E ) = max  , f t R (st )dst f (s )dst E tL t   E Pr(st ∈ E | ω = L ) Pr(st ∈ E | ω = R ) . = max , Pr(st ∈ E | ω = R ) Pr(st ∈ E | ω = L )

If the DM does not know the realization of st but only that st ∈ E, the larger more probable that the true state is L.

 E

f t L (st )dst compared to

 E

f t R (st )dst , the

Preliminary information of signals Upon receiving a signal st and before the DM decides whether to process or ignore it, he observes some preliminary information about its realization. For example, in the context of Google news page, the preliminary information are the titles of articles, graphics, sources, dates of publication and previews.12 This information allows the DM to get a grasp on the content of the article with negligible time and attention, and then to decide whether to carefully read the article based on his preliminary understanding of its content. Formally, the preliminary information is represented by a signal of the signal st , which is denoted by s˜t . I assume that s˜t is drawn from a set E and with a p.d.f. h(· | st ) : (0, ∞) → E . For example, the DM could observe whether the signal 8

The results hold for arbitrarily big T . This is implied by the Bayesian formula. 10 For example, the results on preference for strong signals, confirmation bias for confident individuals and wishful thinking hold for discrete distributions (as in the illustrative example), and for distributions with mass points, assuming in each period there exists strong enough belief-challenging signals in the support of f t to convince the DM to take action r. 11 Note that the definition of belief-confirming and belief-challenging information corresponds to whether the information supports the a priori optimal action. Therefore, it depends not only on the belief of the DM, but also on the utility matrix. In particular, the set of belief-confirming and belief-challenging information may differ for two individuals who have same the beliefs but different utility functions. 12 In general, the preliminary information could be interpreted as a brief understanding of the content of the article that consume negligible amount of cognitive resources. 9

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351

Fig. 1. Timeline of the game at any period t ≤ T .

supports state L or state R but not its strength. In that case, E = {−, ±, +} with s˜t equals + if st > 1, ± if st = 1, − if st < 1. I denote by g (· | s˜t ) the conditional probability distribution of st given the preliminary information s˜t . In the rest of the paper, without loss of generality, I often use g instead of s˜t to denote the preliminary information, as it summarizes the knowledge of the DM about the realization of st . Note that in the illustrative example, g is degenerate, i.e., the DM observes perfectly the realization of st before taking the processing decision. Processing constraint and timeline The DM is subject to a capacity constraint, in which he can only “process” up to T¯ signals. If the DM processes a signal, he learns perfectly13 the realization of st and updates his belief: he reads the articles, understands better the content and incroporates the information into his belief. Otherwise, he ignores the signal without updating his belief. This mechanism of “processing” implies that the DM updates his belief only with the processed signals. Mathematically, let the set of processed signals at the beginning of period t as Mt , which is also referred as the DM’s memory.14 By construction, M1 = ∅. If the DM processes st , Mt +1 = Mt ∪ {st }; otherwise, Mt +1 = Mt . With some abuse of notations, I write t ∈ Mt for some t < t if and only if the DM has processed the signal in period t , while t ∈ / Mt is defined analogously. The belief of the DM at period t, denoted ( ptL (Mt ), ptR (Mt )), is a function of his memory Mt and his prior, which will be defined formally later in the section. The timeline of the game is shown in Fig. 1. In period t = 1, · · · , T , the DM observes the preliminary information s˜t and decides whether to process or ignore the signal. Once the DM has exhausted his processing capacity, the game jumps to period T + 1 and he takes one of the two actions. For simplicity, I assume that he cannot return to a piece of information he has ignored.15 Apart from being subject to the processing constraint, the DM is rational and able to evaluate the expected utility of processing and ignoring a signal, using his memory Mt and the preliminary information g. The processing strategy of the DM at period t is denoted by σt ( g , Mt ) ∈ { P , I }, where P represents the decision to process the signal and I represents the decision to ignore it. The processing strategy from period 1 to period T is denoted by σ = (σt )tT=1 . I assume that if the DM is indifferent between processing or ignoring a signal, he processes it when the signal is belief-confirming and ignores it otherwise. Posterior belief and equilibrium concept I consider an “aware” and an “unaware” case on how the DM forms belief with his memory. The two cases correspond to two different levels of cognitive sophistication of the DM and are defined as follows. More specifically, the “unaware” case corresponds to the simplifying assumption made in the illustrative example, while the “aware” case relaxes the assumption. Unaware DM In the unaware case, when the DM forms his belief, he does not infer information from the fact that he had chosen to ignore some previous signals. As a result, he updates his belief as if he has only received signals in his memory. Formally, the posterior belief of the unaware DM at period t satisfies:

ptL (Mt ) = ptR (Mt )

pL



t ∈Mt

pL



t ∈Mt

f t L (st )

f t L (st ) + p R



t ∈Mt

f t R (st )

(1)

= 1 − Pr L (Mt ).

This unawareness in belief formation has been documented in the literature, as shown in an experimental study conducted by Enke and Zimmermann (2017). They show that a significant amount of their subjects are not aware of, and do not take into account the correlation between different pieces of information when they form beliefs. Analogously, the unaware DM

13

This effect is absent in the illustrative example, where g is degenerate. This corresponds to one of the interpretation of processing, which is to memorize the information in order to update one’s belief. 15 This assumption rules out the possibility that the DM keeps in mind the preliminary information of several pieces of news at the same time and decides which one(s) to process. It emphasizes the limitation of the DM’s cognitive ability and the fact that the capacity of our working memory is small. 14

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is not aware of, and does not take into account his previous processing decisions when he forms belief. This assumption is also equivalent to the equilibrium concepts introduced by Jehiel (2005) and Eyster and Rabin (2005), in which they argue that individuals may not take into account how other people’s action depend on their own information.16 The game is a multi-selfs dynamic game. With the “unawareness” assumption, the DM acts as if there is no incomplete information about the signals he had received and ignored.17 The solution concept used is subgame perfect Nash equilibrium. At each period t = 1, · · · , T , he anticipates how his processing decision affects the decisions of all his future selfs, using his belief of the state ptL (Mt ) and the preliminary information g of the signal st . On the other hand, in period T + 1,

the DM takes actions l if p TL +1 (M T +1 ) > 1/2 and r if p TL +1 (M T +1 ) < 1/2. In case of indifference, I assume that the DM takes action l. Given the model is a single-player game, I will also refer the equilibrium processing strategy as an optimal processing strategy. In the main text, I study the DM’s processing behavior in only the unaware case. However, to help the reader to understand the differences between the two cases, I now briefly introduce the definition of the aware case18 (the formal definition and analysis are presented in the online appendix). The results in the two cases are similar. Aware DM In the aware case, the DM is more sophisticated. When he updates his belief with his memory, he rationally infers information about the signals in periods {t | t ∈ / Mt } from the fact that he had chosen to ignore them. The game is a multi-selfs dynamic game with incomplete information and the corresponding solution concept is perfect Bayesian Nash equilibrium. When the DM updates his belief with Mt , he does not know his processing strategy in periods t < t, but forms a 1 1 ˜ t )tt − conjecture denoted by (σ˜ t )tt − =1 . In equilibrium, for all t = 1, · · · , T + 1, the conjecture (σ =1 of the DM’s period t self has to be consistent with the equilibrium processing strategies of his previous selfs. On the other hand, when the DM decides whether to process or ignore a signal, he is fully aware of the fact that his future self will make rational inference. He evaluates the expected utility of processing and ignoring a signal and takes the optimal processing decision, using his knowledge of the state and the corresponding signal realization. 4.1. Discussion Before I present the analysis, I discuss the assumptions of the model. The two core building blocks of the model are the DM’s inability to process all available signals and his ability to make optimal processing decisions based on his understanding of the signals realizations. While the first models the limitation in cognitive ability, the second imposes a high cognitive demand on the DM as he has to evaluate the probability of the states and anticipate how processing or ignoring the signals affects the decisions of his future selfs. Thus, this paper tries to take a minimal departure from the literature (the first building block) while keeping the rational benchmark (the second building block) to highlight how the capacity constraint can drive selective/biased processing behavior. In the following, I provide several justifications/interpretations of the setting. 1. Short-term-long-term-memory conversion The first interpretation of the model is a memory-based belief formation mechanism.19 The DM forms belief only with his (long-term) memory of information. Upon receiving a piece of information, it first enters as a short-term memory. The DM then evaluates (consciously or unconsciously) the benefits and costs of processing, and decides whether or not to consume his cognitive resources to convert it into long-term memory. 2. Title (or a brief look) and the main text of articles The second interpretation is related to the way individuals process news articles. The DM first reads the title of the article and roughly gets an impression of its main message. He then decides whether or not to read the main text based on this rough impression, which allows him to update his belief accordingly. If he chooses not to read the main text, his rough impression of the article has negligible effect on his belief. 3. Optimal selective processing Taking a step back, there is abundant evidence that individuals update differently with belief-confirming and beliefchallenging information. This model studies the characteristics of the optimal selective processing strategies given the DM’s limited processing ability, and analyzes how it resembles different empirical phenomena and how it changes with the informational environment.

16 In the current setting, when the unaware DM forms his belief, he does not take into account the fact that his previous selfs’ processing decisions depend on their knowledge of the realization of the signals. As a result, he forms his belief as if the processing decisions of his previous selfs did not correlate with the signals’ realizations. 17 The DM acts as if he had only received the signals in his memory. In other words, it is as if he “mistakenly thinks” that he is in a singleton information set. 18 Note that both aware and unaware case are models with imperfect recall (Piccione and Rubinstein (1997)), i.e., the DM does not recall the signals he once received but chose to ignore, but the two cases differ in the DM’s awareness of his absentmindedness. In particular, the unaware DM forms belief as if he has perfect recall. 19 See section 2 for related literature.

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4. Delegation on information acquisition Although the aim of this paper is to analyze individual behavior, the model could be interpreted as a game between a decision maker and multiple delegates who acquire and present information. Imagine T delegates who share the same preference as the decision maker. Each of the delegate acquires one piece of information and sequentially decides whether to present it publicly (in front of other delegates) to the decision maker, who will take an action based on the presented information. The model depicts the situation where there is a capacity constraint on how many signals the delegates could present to the decision maker, which could be imposed by cost/time concerns or organization protocols. 5. Processing strategy when T¯ > 1 In this section, I characterize the processing strategy of the DM in period t, given preliminary information g. I first analyze the case where g is degenerate, i.e., the preliminary information perfectly reveals the signal realization, and then proceed to discuss the differences with non-degenerate g. A degenerate distribution g is simply denoted by st . For simplicity of notations, I use ( ptL , ptR ) to denote the belief of the DM at the beginning of period t, where I assume ptL u L ≥ ptR u R without loss of generality. To evaluate the expected utility of processing and ignoring a signal st , the DM has to anticipate how his processing +1 decision will affect the action chosen by his future self. To simplify expressions, denote by PrtM (a | ω) the probability that t +1 the DM takes action a in state ω , evaluated at the beginning of period t + 1 given his memory Mt +1 . The DM’s expected utility of processing st , denoted by U P (st , Mt ) is equal to the probability of matching his action with the state, weighted by the payoff u L , u R :

U P (st , Mt ) = u L

ptL st ptL st + ptR

ptR

+1 PrtM (l | L ) + u R t ∪{st }

ptL st + ptR

+1 PrtM (r | R ). t ∪{st }

(2)

ptL st

+1 is the conditional probability of state L given the signal st while PrtM (l | L ) is the probability t ∪{st } that his future self successfully matches his action with state L given that Mt +1 = Mt ∪ {st }. Similarly, his expected utility after ignoring the signal, denoted by U I (st , Mt ), follows:

In the first term,

ptL st + ptR

U I (st , Mt ) = u L

ptL st ptL st + ptR

+1 PrtM (l | L ) + u R t

ptR ptR + ptL st

+1 PrtM (r | R ). t

(3)

By comparing equation (2) and (3), we could see that the processing decision affects the utility as it induces a different distribution on the action chosen by his future self. Denote by ts ,M (a | ω) the change in probability of choosing action a t t in state ω after processing st , i.e., +1 +1 tst ,Mt (a | ω) = PrtM (a | ω) − PrtM (a | ω), t ∪{st } t

we have U P (st , Mt ) ≥ U I (st , Mt ) if and only if:

uL

⇐⇒

ptL st ptL st + ptR

tst ,Mt (l | L ) ≥ −u R

ptR ptR + ptL st

tst ,Mt (r | R )

u L ptL st tst ,Mt (l | L ) ≥ −u R ptR tst ,Mt (r | R ).

(4)

Equation (4) shows how the DM’s processing decision depends on the change in the quality of decision making in the two states. For instance, if both ts ,M (l | L ) and ts ,M (r | R ) are positive, processing the signal improves the quality of t

t

t

t

decision making in both states, i.e., there is a higher probability that the DM chooses the optimal action in both states.20 Unsurprisingly, by equation (4), the DM processes the signal. The opposite holds true when both ts ,M (l | L ) and ts ,M (r | t

t

t

t

R ) are negative.21 The trade-off kicks in when the two functions have opposite signs, i.e., processing the signal improves the quality of decision making in one state but worsens it in another state. The following lemma characterizes the DM’s processing strategy given ts ,M (l | L ) and ts ,M (r | R ) are of opposite signs. t

t

t

t

Lemma 1. Assume ts ,M (l | L ) and ts ,M (r | R ) are of opposite signs with neither equal to 0, and suppose the DM is not indifferent t t t t between processing and ignoring. At period t, the DM’s processing strategy is as follows:

20 Note that this could be possible only because processing a signal could affect the processing strategies of the DM’s future selfs. For example, suppose processing a signal induces his future selfs to be more selective that he only processes stronger signals. If there are enough cognitive resources and available signals such that there is a high probability that he indeed receive stronger signals, then it could improve his decision making in both states as it is now based on stronger signals. 21 With limited attention, processing a signal does not necessarily increase the probability of choosing the optimal action in either state. It is because processing lowers the number of signals the DM can process in the future. Take an extreme example where st = 1, processing the signal does not help the DM to know better about the state of the world, but consumes one unit of processing capacity that could be used on an informative future signal.

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1. if ts ,M (l | L ) > 0, the DM chooses to process it if and only if22 t t

st >

u R ptR u L ptL

×

tst ,Mt (r | R ) tst ,Mt (r | L )

;

(5)

2. if ts ,M (r | R ) > 0, the DM chooses to process it if and only if t t

st−1 >

u L ptL u R ptR

×

tst ,Mt (l | L ) tst ,Mt (l | R )

(6)

.

To understand the lemma, let us focus on case 1 where ts ,M (l | L ) > 0. Note that the numerator of the term on t t the right-hand-side of equation (5) measures the loss in utility in state R while the denominator measures the gain in utility in state L, with both weighted by the prior beliefs. Thus the R.H.S. measures the loss-to-gain ratio of processing the signal st , adjusted with the prior belief. If st is big enough, the signal strongly suggests that state L is true and thus the probability of incurring the loss is small, which gives the DM incentive to process. Similar intuition applies for case 2 where tst ,Mt (r | R ) > 0. Using Lemma 1 and to shed light on the role of the capacity constraint, the following proposition, which applies to both degenerate and non-degenerate g, compares the equilibrium processing strategies of the DM without and with limited cognitive ability, i.e., where T¯ ≥ T and T¯ < T .



Proposition 1. The DM processes all informative signals g in equilibrium, i.e., s=1 dG (s) > 0, if and only if T¯ ≥ T (with exception in period T where he is indifferent between processing or ignoring signals that will not change his action). Without limited cognitive ability, there is no trade off between current and future signals. Unsurprisingly, it is always optimal to process all informative signals, as otherwise the DM will have “incorrect” belief about the state and may take an suboptimal action. More interestingly, the trade off between current and future signals kicks in when T¯ < T , such that the DM ignores informative signals with positive probability. Suppose to the contrary of Proposition 1, the DM processes all signals he receives. At period t, the decision to process or ignore st is equivalent to the decision on whether to take action based on (st , · · · st + T¯ −1 ) or on (st +1 , · · · st + T¯ ), i.e., the DM trades off between st and st + T¯ . When st is very uninformative (close to 1), the gain of processing is small. It is optimal for the DM to save his processing capacity for st + T¯ , which is in average more informative than st ≈ 1. Apart from strategically ignoring signals, the DM’s processing strategy is “asymmetric”. The two inequalities (5) and (6), which characterize the processing decisions regarding to signals that increase the probability of choosing action l and r respectively, are generically different. This asymmetry drives differences between the processing behavior regarding information supporting or against the a priori optimal action. As will be explored in more details in next section, this asymmetry in processing strategy explains a number of biases documented in the literature, including confirmation bias for confident individuals. Non-degenerate g Now I turn to the case where g is non-degenerate, i.e., the preliminary information g does not perfectly reveal the realization of the signal st . In this case, processing the signal has two effects: it allows the DM to learn the realization of st and update his belief. The expected utilities of processing and ignoring a signal with preliminary information g are:



U P ( g , Mt ) = U I ( g , Mt ) =



U P (st , Mt ) g (st | s˜t ) dst ; U I (st , Mt ) g (st | s˜t ) dst ,

which gives the following result. Corollary 1. Suppose when g is degenerate, the DM’s processing strategy at period t is to process st if and only if its realization is in some set S¯ , i.e., σt (st , Mt ) = P if and only if st ∈ S¯ . Fix (σt )tT =t +1 and now suppose the DM observes that st ∼ λ × g S¯ + (1 − λ) × g S¯ C for some λ ∈ [0, 1] and g S¯ , g S¯ C where

¯ < 1 such supp( g S¯ ) ⊆ S¯ and supp( g S¯ C ) ⊆ S¯ C . He processes that signal if and only if λ is big enough, i.e., fixed g S¯ and g S¯ C , ∃ λ ¯ and σt ( g , Mt ) = I if λ < λ¯ . that σt ( g , Mt ) = P if λ > λ 22

Note that tst ,Mt (r | ω) = −tst ,Mt (l | ω).

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Fig. 2. The DM processes s1 if its realization is inside the shaded region.

Importantly, Corollary 1 implies that one could focus on the case where g is degenerate, without losing much qualitative insights. For instance, if the DM processes belief-confirming signals but ignores belief-challenging signals, he also processes a signal if he observes that the signal is likely to be belief-confirming. In the rest of the paper (apart from Corollary 2), I analyze only the case where g is degenerate. 6. Continuation game where T¯ = 1 In this section, to show more detailed behavioral implications, I present the analysis of the continuation game where T¯ = 1, i.e., the DM can only process one signal. Furthermore, I will only look into the case where T = 2, i.e., there are only two available signals. In the online appendix, I show that it is without loss of generality23 to do so when T¯ = 1. I am thus interested in the processing decision in period 1, i.e., whether the DM processes s1 or ignores s1 in order to save his cognitive resources for s2 . Using24 Lemma 1 and remember that I focus on the case where g is degenerate, the equilibrium processing strategy of the DM is characterized in the following proposition. Proposition 2. In period 1, the DM processes a signal s1 if and only if it is strong enough, i.e.,

• he ignores pure noise s1 = 1; • he processes a belief-confirming signal s1 > 1 if and only if STR (s1 ) = s1 ≥ + ;

(7)

• he processes a belief-challenging signal s1 < 1 if and only if25 1 − STR (s1 ) = s− 1 > ,

(8)

where + and − are defined as follows:

  u p  R R × STR s2 ∈ 0, ; uL pL uL pL   u p  uL pL uL pR R R − = × STR s2 ∈ ,∞ > .

+ =

uR pR

uR pR

uL pL

(9)

uR pR

The equilibrium processing strategy of the DM is a threshold-strategy, i.e., he processes the signal if and only if the signal strength is larger than some threshold as illustrated in Fig. 2. As shown by equation (9), the two thresholds are generically different, i.e., the processing strategy is asymmetric as pointed out in the previous section. Moreover, the loss-to-gain ratio of processing s1 is captured by the strength of a set of “contradictory” future signal s2 , which corresponds to the realizations that induce a different action compared to s1 (see Table 3). For example, consider s1 > u p 1. The bigger F 2R ( uR p R ) is, the higher the probability the DM will receive contradictory future signal in state R and switch L

L

u p

(optimally) to action r; the smaller F 2L ( uR p R ) is, the less likely the DM will receive contradictory future signal in state L L L and switch (sub-optimally) to action r. Therefore, the stronger the contradictory signals are, the bigger the loss-to-gain ratio of processing s1 (or forgoing s2 ) is. Loosely speaking, when an individual decides whether not to read a piece of left-wing news, he is concerned about losing time and/or attention for a right-leaning article as it provides contradictory information. Thus, the stronger the contradictory future signals are compared to the current signal, the more incentive the DM has to ignore the current signal and save his cognitive resources for the future signal. This feature has important implications on the comparative analysis, which will be discussed later in this section.

23 Loosely speaking, I show that the equilibrium processing strategy of the DM in any period t < T is equivalent to the equilibrium processing strategy of the DM in period 1 in a game where T = 2 with some information structure ˜f 2L , ˜f 2R . 24 When T¯ = 1, 1s1 ,M1 (l | L ) and 1s1 ,M1 (r | R ) are always of opposite signs. 25 Note that inequality (7) is weak while inequality (8) is strong because of the tie-breaking assumption: the DM processes belief-confirming signals but ignores belief-challenging signals when he is indifferent.

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Table 3 The table shows the final action given (s1 , s2 ) and the processing decision. The colored boxes highlight the configurations of s2 that induce different actions compared to s1 , which corresponds to the subset of realizations of s2 shown in equation (9). Final action s1

≥

<

s2 uR pR uL pL

uR pR uL pL

≥ < ≥ <

process s1 uR uL uR uL uR uL uR uL

pR pL pR pL pR pL pR pL

ignore s1

l

l

l

r

r

l

r

r

6.1. Behavioral phenomena The equilibrium processing strategy has behavioral implications which explain some of the biased processing behaviors documented in the behavioral economics and psychological literature. The first implication is that there is a preference for strong signals, i.e., the DM processes only strong enough signals in equilibrium. This implies that individuals react more to striking or convincing information but ignore coarse or ambiguous information. Analogously, in an experiment which studies visual attention, Itti and Baldi (2006) shows that individuals selectively allocate visual attention to details that induce a large difference between prior and posterior belief. Another implication relates to the confirmation bias, which is defined as follows: Definition 1 (Confirmation bias). The equilibrium processing strategy exhibits confirmation bias if − > + . That is, the DM processes a larger set of belief-confirming information than belief-challenging information. When − > + , the DM processes a belief-confirming signal s but ignores an equally strong belief-challenging signal s−1 for all s ∈ [+ , − ]. On the other hand, there does not exist any s ∈ (1, ∞) such that the DM ignores a belief-confirming signal s but processes an equally strong belief-challenging signal s−1 . The following proposition presents two behavioral implications of the equilibrium processing strategy. Proposition 3. The equilibrium processing strategy of the DM explains the following behavioral phenomena26: 1. (preference for strong signals) The DM processes a signal if and only if it is strong enough, as shown in equation (7) and (8). 2. (confirmation bias for confident individuals) Fix27 f 2L , f 2R , u L and u R , there exists p¯ < 1 such that for all p L ≥ p¯ , the optimal processing strategy exhibits confirmation bias, i.e., − > + . The intuition of Proposition 3 can be understood as a trade-off between current and future information. On one hand, processing the current signal induces a loss as the DM forgoes informative future signal; on the other hand, it induces a gain as otherwise he will not take the current signal into account when he takes action l or r. First, the stronger (more informative) the current signal is, the more important it is to take it into account in the decision making problem so the gain is bigger. A stronger current signal also implies a more extreme posterior and the future information becomes less important. Both drives the preference for strong signals. Second, when the DM is a priori confident enough that state L is true, even if he receives a weak belief-confirming signal, he becomes almost sure that action l is the optimal action. Future information becomes less important as there is u p a small probability of receiving a very strong contradictory signal s2 < u Rp Rs . The gain of processing s1 , to prevent himself u p

u p

L

L 1

from switching sub-optimally when s2 ∈ [ u Rp Rs , uR p R ), dominates the loss when p L is big enough. In contrast, even when L L 1 L L the DM has processed a strong belief-challenging signal, he is not sure about the optimality of action r. Therefore, the future information becomes important, which induces the DM to ignore the current signal and save the capacity for the

26 Note that the ignorance of weak signals, implying stickiness of belief, is in reminiscent of the stickiness of prices in Sims (2003), while confirmation bias relates to whether belief is sticky-up or stick-down. 27 Note that as signals are normalized as their odds ratio, f 1L and f 1R do not affect the equilibrium processing strategy.

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future signal. This asymmetry28 in the incentive to look for future information drives the confirmation bias for confident individuals. The result of confirmation bias also holds when g is non-degenerate. To illustrate that, consider two pieces of preliminary information, g + and g − , where

g + (s) = 0 for all s ≤ 1; g − (s) = 0 for all s ≥ 1. That is, the DM observes that the signal s1 is belief-confirming if s1 ∼ g + and is belief-challenging if s1 ∼ g − . In both cases, he is not sure about the signal strength. Corollary 2. Fix f 2L , f 2R , u L and u R , there exists p¯ < 1 such that for all p L ≥ p¯ , i.e., when the DM is a priori confident enough, he processes the signal in period 1 if he observes that s1 ∼ g + but ignores the signal if he observes that s1 ∼ g − . Both Proposition 3 and Corollary 2 explain the phenomenon of information avoidance, in particular the avoidance of belief-challenging information (Kahan et al. (2012)). Although this behavior of information avoidance looks like a systematic bias against belief-challenging information, the results aforementioned suggest that when one takes into account the limitation in cognitive ability, confirmation bias actually arises as an optimal strategy under a setting with rational and Bayesian individuals. This is in particular different from the literature which explains the processing behavior with belief-dependent utility, exogenous biases such as time-inconsistent preference or interpersonal interaction.29 Moreover, as the optimal processing strategy may not exhibit confirmation bias when p L is close to 12 , the results also help us to understand how the bias differs across individuals/subjects with different prior beliefs. It is interesting to note the limitation in processing ability drives not only a confirmation bias, but also a bias in action. More specifically, if the DM is a priori confident enough, he chooses action l with a higher probability in the current setting, than in a setting where there is no limitation in processing ability. Corollary 3 (Bias in action for confident individuals). Fix f 2L , f 2R , u L and u R , there exists p¯ < 1 such that for all p L ≥ p¯ , i.e., when the DM is a priori confident enough, he chooses action l with a higher probability when he can only process one of the two signals s1 and s2 , than when he can process both. Now I present another behavioral implication, which relates to wishful thinking. It refers to individuals’ tendency to form optimistic belief about some desirable outcomes. In the psychological theory proposed by Krizan and Windschitl (2007), one of the mechanisms driving wishful thinking is that individuals tend to process and encode information that suggests some desirable outcomes. In the current setting, u L and u R measures the how desirable the two states are. When u L is larger than u R , state L is more desirable than state R as it is associated with a higher achievable payoff. By analogy to confirmation bias, wishful thinking is defined as follows: Definition 2 (Wishful thinking). Suppose state L is more desirable than state R, i.e., u L ≥ u R . The equilibrium processing strategy of the DM exhibits wishful thinking if − > + . That is, he processes a larger set of information that supports state L, compared to that supports state R. It is clear from equation (7) and (8) that an increase in u L /u R has the same effect as an increase in p L / p R . Therefore, the second point of Proposition 3 implies the following result. Corollary 4 (Wishful thinking). Fix f 2L , f 2R and p L , and without loss of generality assume u L ≥ u R , there exists u¯ ∈ [1, ∞) such ¯ i.e., state L is much more desirable than state R, the equilibrium processing strategy of the DM exhibits wishful that when u L /u R ≥ u, thinking, i.e., − > + .30 6.2. Comparative analysis I now analyze how a change in the information structures ( f 2L , f 2R ), which could be induced by the Internet or information policies, affects the information processing behavior of the DM. In particular, I study whether the change strengthens the confirmation bias. Consider two environments A and B, which are associated with two different equilibrium processing

28

Note that the gain of processing always dominates the loss when the DM are almost sure about which action is optimal, i.e., when

u L p L s1 uR pR 29

→ ∞; the reverse is always true when the DM is indecisive about the two action, i.e., when

u L p L s1 uR pR

≈ 1.

u L p L s1 uR pR

→ 0 or

See section 2 for the references. Note that the model also explains a reverse phenomenon of wishful thinking, namely that individuals tend to form pessimistic belief about some undesirable outcomes when the associated loss increases (Dunning and Balcetis (2013)). The formal result is presented as Corollary 9 in the appendix. 30

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strategies characterized by two sets of thresholds (+ , − ), i = A , B. The confirmation biases under the two environments i i are compared as follows: Definition 3 (Comparison of confirmation bias). The confirmation bias is stronger under environment A than under environ+ − − ment B if + A ≤  B and  A ≥  B , where at least one of the two inequalities is strict. That is, under environment A, the DM processes only a subset of belief-challenging information that he would process under environment B; while the reverse is true for belief-confirming information. In the sequel, I analyze two specific types of change in the information structure, in order to illustrate how the Internet could strengthen the confirmation bias of the DM. The first type of change captures the idea that the Internet facilitates a better access to information, thanks to the decrease in information transmission cost and the advance of search technology. Formally, I assume that under environment j ∈ { A , B }, the signal s2 is drawn from the following distribution31 :



s2 =

1

with probability 1 − λ j ;

s

otherwise, where s ∼ ( f 2L , f 2R ).

The bigger λ j is, the better the access to information is. When λ A > λ B , there is a higher probability to receive informative signals s2 = 1 under environment A than that under environment B. Proposition 4. If environment A facilitates a better access to information compared to environment B, the confirmation bias is stronger + − − under environment A than under environment B, i.e., if λ A > λ B , we have + A =  B and  A >  B . The result is driven by the asymmetric effect on the set of contradictory future signals when the current signal is belief-confirming or belief-challenging respectively. More specifically, a better access to information strengthens the set of u p u p contradictory signals (s2 ∈ [ uR p R , ∞)) for s1 < 1, but has no effect on the set of contradictory signal (s2 ∈ (0, uR p R )) for L L L L s1 > 1, and that incentivizes the DM to ignores belief-challenging signals. Before I further comment on the result, let me first present the analysis on the second type of change. It corresponds to the idea that the Internet lowers the barrier to information production. As a result, it facilitates the development of new media, and therefore provides individuals with more information sources. The quality of the new information sources could be different from the old ones. Intuitively, the skewness of the information structure of the new sources have a great impact on the processing strategy of the DM. For instance, if the new information sources produce strong belief-challenging information but weak belief-confirming information, the DM has more incentive to process the former and ignore the latter. To avoid assuming whether the new information sources are biased towards one state or the other, I analyze the case where the information structure of the new sources are in aggregate symmetric. Formally, I consider the following extension of the first type of change. Under environment B, the signal s2 is drawn from the following distribution:



s2 =

1

with probability 1 − λ

sB

B B otherwise, where s B ∼ ( f 2L , f 2R ),

(10)

while under environment A, the signal s2 is drawn from the following distribution:

s2 =

⎧ ⎪ ⎨1 ⎪ ⎩

with probability 1 − λ − δ B B with probability λ, where s B ∼ ( f 2L , f 2R );

sB

with probability δ , where s A ∼

sA

(11)

A A ( f 2L , f 2R )

A A A A for some symmetric distribution ( f 2L , f 2R ), i.e., f 2L (s) = f 2R (s−1 ). Put differently, environment A transfers some mass δ of A A pure noise to the new information sources, which is characterized by a symmetric distribution ( f 2L , f 2R ).

− Proposition 5. Consider two environments, A and B, defined in equation (11) and (10) respectively, we have − A >  B for all + + A A B B ( f 2L , f 2R ) and ( f 2L , f 2R ), while  A ≤  B if and only if





STR s A ∈ −∞,



uR pR  uL pL

   uR pR  ≤ STR s B ∈ −∞, . uL pL

That is, the confirmation bias is stronger under environment A compared to environment B if and only if the strong belief-challenging information drawn from the new sources is weaker than the same information drawn from the old information sources.

31

Note that unlike in previous sections, the c.d.f.s. of s2 defined here are not continuous. Nonetheless, the results in previous sections still hold.

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Table 4 The distribution of st , t = 1, 2, given the state of the world.

ω=L ω=R

f t ω (1/q)

f t ω (1)

f t ω (q)

λ(1 + q)−1

1−λ

λq(1 + q)−1

λq(1 + q)−1

1−λ

λ(1 + q)−1

Note that both types of change are defined in a way such that the information structure under environment A is in average more informative32 than that under environment B. When the future information becomes more informative, intuition suggests that the DM should have more incentive to ignore the current information and process the future information. However, in this model, the processing decision depends not on the average informativeness of all realizations of s2 , but only on the set of contradictory realizations of s2 . For instance, when the DM decides whether to process a left leaning article, he does not worry about losing attention for another left leaning article but worry about losing attention for a right leaning article. While environment A increases the average strength of s2 compared to environment B, it does not necesu p sarily increase the average strength of the strong belief-challenging information s2 < uR p R . As a result, providing in average L L better information to the DM could strengthen his confirmation bias. The Internet undoubtedly facilitates better access to information and lowers the cost of information production. As the Internet lowers the entry cost of information production, the new information sources are in general of lower quality. Proposition 4 and 5 contribute theoretical explanations and insights to the literature of political polarization, as it suggests that the Internet promotes biased processing behaviors or so called “cherry-picking” on information, in the presence of information overload. 7. Applications In this section, I provide two simple examples, which relates to polarization and media competition in the presence of information overload. For simplicity, I only look at the case where u L = u R = 1, T = 2 and T¯ = 1. Moreover, as in the illustrative example, I assume s1 and s2 follow a symmetric information structure with three possible realizations, denoted as 1/q, 1, q where q > 1. With some abuse of notation, I use f t ω to denote the discrete distribution in state ω ∈ { L , R } and period t = 1, 2, which is represented in the following table: An increase in λ represents a better access to valuable information while q represents the strength/quality of the informative signals. In the illustrative example, λ = 0.3 and q = 2.5. Throughout the section, I focus on the unaware case and assume g is degenerate. 7.1. Polarization In the following, I show how the model of limited cognitive ability presented in this paper sheds light on the wide range of phenomena documented in the literature of political polarization (see section 2). Alice and Bob have opposite prior beliefs about the state of the world with obvious notations paL = p bR = p > 1/2. They receive the same two (public) signals p drawn from the distribution shown in Table 4. I assume that the informative signals are strong enough, i.e., q > 1− p , which rules out the trivial case where the two individuals always take their a priori optimal action. Without loss of generality, I assume that L is the true state. I analyze three indicators of polarization: 1. the probability that both individuals take the same action, which is denoted as P consensus . It is used to assess the intuition where more/better information results in a higher probability of achieving consensus; j 2. the probability that the individual takes his/her a priori optimal action, which is denoted as P default , j = A , B. It measures how well the a priori optimal action, or analogously party identification, predicts voting decision, which is studied in the empirical literature of political science (e.g., Bartels (1998) and Bartels (2000)); 3. the change in the distance between the beliefs of the two individuals after receiving information. It corresponds to belief polarization and is widely analyzed in the theoretical economics literature.33 Note that the three indicators study short/medium-run polarization when individuals receive the same public information, instead of long-run disagreement.34 Using Proposition 2, the processing strategy of the two individuals are characterized by a set of thresholds q− (λ, p ), λ− (q, p ) and p − (λ, q). For simplicity of exposition, I omit the arguments of the thresholds for the rest of the paper.

32

The average strength of signal s2 in environment A is bigger than that in environment B, i.e., E A (STR(s2 )) > E B (STR(s2 )). For example, see Baldassarri and Gelman (2008), Acemoglu et al. (2007). 34 For example, if the signal space spans R++ such that there are arbitrarily strong signals supporting the two states, when T goes to infinity, there won’t be disagreement in the long run. As there is no discounting, individuals will almost learn the true state as they can wait for arbitrarily strong signals. 33

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Fig. 3. Alice’s processing strategy in period 1 as a function of λ and q, fixing paL = p = 0.7.

Fig. 4. The effect of λ on the first two indicators of polarization.

Corollary 5. In period 1, both Alice and Bob process s1 if it is belief-confirming and ignore it if it is pure noise. There exists some thresholds q− , λ− and p − such that they process belief-challenging signal if and only if

• fixing q and λ, their priors are weak enough, i.e., p < p − , or; • fixing p and λ, the informative signals are strong enough, i.e., q > q− , or; • fixing q and p, the access to information is poor enough, i.e., λ < λ− . The proof is shown in the appendix and the result is illustrated in Fig. 3. The first two points correspond to the result of “confirmation bias for confident individuals” and “preference for strong signals”, while the third point corresponds to Proposition 4 that better access to information strengthens confirmation bias. The following proposition, illustrated in Fig. 4, j presents how a change in the access to information λ affects the first two indicators of polarization, P consensus and P de f ault . Proposition 6. The probability that Alice and Bob take the same action, P consensus , is non-monotonic in λ, i.e., it is increasing in the range [0, λ− ) and [λ− , 1], but exhibits a downward jump at λ = λ− . a b Similarly, the probability that Alice/Bob takes her/his a priori optimal action, P de or P de , is also non-monotonic in λ, i.e., f ault f ault both are decreasing in λ in the range [0, λ− ), but exhibit a upward jump at λ = λ− .

When there is better access to information, fixing the processing strategies of the two individuals, the probability that they take the optimal action l increases. However, as shown in Proposition 4 and Corollary 5, a better access to information promotes biased processing behavior, i.e., when λ is big enough, the two individuals ignore belief-challenging information. As a result, it could reduce the probability of achieving consensus and increase the probability that the individuals take their a priori optimal action, despite the availability of more valuable public information (Gentzkow and Shapiro (2011) and Flaxman et al. (2016)). The limitation in processing ability hinders the benefits of information technology because individuals strategically allocate their cognitive resources in the presence of information overload. Moreover, as individuals with different prior beliefs adopt different processing strategies, their beliefs can be polarized even if they receive the same sequence of information. Corollary 6. When λ ≥ λ− , q ≤ q− or p ≥ p − , the distance between the beliefs of Alice and Bob increases after receiving information if the signals in the two periods support different states. More specifically, if s1 > 1 > s2 or s2 > 1 > s1 , Alice becomes more confident that state L is true while Bob becomes more confident that state R is true. Moreover, the probability of such belief polarization increases in λ and decreases in q.

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Corollary 6 shows that even when the same sequence of information is available for both individuals, the difference in their prior beliefs induces different processing strategies and could polarize their beliefs. Moreover, it happens only when there are sufficiently good access to information, or when the prior beliefs of the two individuals are strong enough. In other words, when the two individuals are partisan enough, a better access to information gives rise to the possibility of polarization even under a setting with public information. This result sheds light on the empirical evidence that polarization is much stronger among individuals who are more partisan (Evans (2003), Baldassarri and Gelman (2008), Abramowitz and Saunders (2008) and Hetherington (2009)). On the other hand, belief polarization happens when the information available in the two periods are contradictory, i.e., s1 > 1 > s2 or s2 > 1 > s1 . Its probability increases when there is better access to information and when the quality of information decreases. Arguably, the Internet contributes to both. While it provides us with enormous amount of information, it also facilitates the spread of rumors, fake news and low-quality information. It is much easier to find contradictory information on the Internet, which gives more incentive for individuals to selectively attend to belief-confirming information and increases the probability of belief polarization. 7.2. Media competition In this second application, I study media strategy in the information era. In order to introduce the role of the media, I present a variation of the main model. Formally, there is a continuum of media, indexed by i ∈ I . In period t = 1, 2, each media i collects a signal sit about the state of the world and publishes a piece of news mit . The signal sit collected by different media are conditionally independent and follow the distribution defined in Table 4. I assume that the media cannot post fake news and therefore either publishes the signal it receives or publishes nothing, i.e., mit ∈ {∅, sit }. To simplify the analysis, I assume that there are three types of non-strategic (biased) media, { T L , T R , T N }, which publish according to the following fixed rules:

• media {i | i ∈ T L } is biased towards state L: it publishes sit if it supports state L (sit = q), but publishes ∅ if it receives sit ∈ {1/q, 1}; • media {i | i ∈ T R } has an opposite bias: it publishes a signal if it supports state R (sit = 1/q) but publishes ∅ if it receives si ∈ {1, q}; • media {i | i ∈ T N } has no bias and publishes any informative signal it receives, i.e., mit = si if and only if si = 1. I assume that each media belongs to one and only one of the three types, and the type of each media is a public information. The time line of the game is as follows: Period 1 The DM chooses which media he visits and processes the piece of news mi1 posted by the media. Period 2 If the media visited by the DM in period 1 published nothing, he chooses again a media outlet to visit. Otherwise, he visits no media as processing the information in period 1 takes time. Period 3 The DM forms his belief with the processed information and takes an action l or r. Note that this variation differs from the main model only in terms of interpretation. Here the DM control his “diet” of information by choosing which (biased or unbiased) media to visit, instead of choosing whether to process or ignore the signals he receives. For example, if the DM visits media i ∈ T L in period 1, it is as if he chooses to process the period 1 signal if and only if it supports state L. By Corollary 5, the DM visits the biased media in period 1 if his prior belief is strong enough. Corollary 7. Fix λ and q, suppose the DM has prior belief ( p L , 1 − p L ) where p L ≥ 1/2. In period 1, there exists a threshold p − ∈ (1/2, 1) such that if p L ≥ p − , he visits a media i where i ∈ T L ; if p L ∈ (1/2, p − ), he visits a media i where i ∈ T N . If the media visited by the DM in period 1 published nothing, he visits a media i where i ∈ T N in period 2. Now I turn to analyze the viewership of the three types of media. In the society, individuals’ prior belief p L are distributed according to g ( p ) where g ( p ) > 0 for all p ∈ (0, 1). Its c.d.f. is denoted by G. Define the viewership of media {i | i ∈ T j } as the mass of individuals which visit media {i | i ∈ T j } across the two periods. Denote it by V j for j = L , R , N. By Corollary 7,

V L = 1 − G ( p − ); V R = G (1 − p − ); V N = 1 − λ( V L + V R ). That is, biased media attracts views from individuals with strong beliefs, while unbiased media serves the others. Corollary 8. When there are better access to information (λ increases), the viewership of the biased media increases while the viewership of unbiased media decreases, i.e., V L and V R increase with λ while V N decreases in λ. On the other hand, when the quality of information increases (q increases), the viewership of the biased media decreases while the viewership of unbiased media increases, i.e. V L and V R decrease with q while V N increases in q.

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2

Proof. The result follows from Corollary 5.

As shown in previous sections, a better access to information or an decrease in quality of information strengthens the confirmation bias of individuals, which in turn increases the viewership of the biased media. The increase in viewership increases the profitability of biased media and thus incentivize media to adopt a biased strategy. This result sheds light on the emergence of partisan media in recent years as the Internet promotes biased processing behavior. 8. Conclusion In conclusion, this paper investigates the information processing behavior of a decision maker who can process only a subset of all available signals. I show that this limitation in processing ability drives a number of well-documented behavioral biases, including preference of strong signals, confirmation bias for confident individuals and wishful thinking. These biases has been attracting lots of attention in the behavioral economics literature, in which many have analyzed how these biases affects different market outcomes by introducing exogenous biases in traditional economics models. In contrast, instead of taking the biases as they are, this paper aims to improve our understanding by analyzing their cause. In particular, I show that these biases are features of optimal strategies if we take into account our limited cognitive ability as a human being. This approach allows us to analyze the biases as an outcome of an optimization problem. It brings two advantages. First, not only that it explains the existence of the biases, but using standard techniques of comparative statistics as illustrated in this paper, it also explains the heterogeneity of the biases among individuals with different personal characteristics or abilities, and how they change in different situations faced by the individuals. Second, it allows us to study how regulatory policies could play a role in changing these biases and associated market outcomes. Thus, looking forward and building from the insights of this paper, there are two different ways to further develop the literature. First, in policy analysis with behavioral settings, modeling the biases as optimal strategies allows us to take into account the indirect effects of policy interventions on behaviors of individuals. It contributes to a more complete analysis than if we take the biases as they are. For example (loosely speaking), if providing more information to consumers strengthens their confirming bias, it might back fire as it could weaken competition and increase prices. Second, more experimental or empirical work has to be done to understand how biases are formed and vary across different individuals or settings. It will give us a better understanding on whether the biases do indeed relate to the limitation in ability. And as an (early) answer to that question, in an experimental study conducted by myself and co-authors, Goette et al. (2019), we find evidences that a larger cognitive load strengthens confirmation bias in belief formation. Appendix A. Omitted results and proofs A.1. Proof of Lemma 1 Proof. The lemma is implied by inequality (4) combined with

tst ,Mt (r | L ) = −st ,Mt (l | L ); tst ,Mt (l | R ) = −st ,Mt (r | R ).

2

A.2. Proof of Proposition 1 Proof. I first prove the “if” statement. First, it is without loss to assume the DM processes all informative signal in period T . In the following, I will focus on the degenerate case and prove that when T¯ ≥ T , and the best response processing strategy is to process st for all st = 1 given that all his future selfs process all st = 1 for t = t + 1, · · · T . Denote the c.d.f. of st +1 st +2 · · · s T in state ω = L , R as F (t +1→ T )ω , i.e.,

∞ F (t +1→ T )ω (s) =



∞ ···

FTω

0

0

∞

∞



s st +1 · · · s T −1

f (t +1→ T ) L (s) =

··· 0

fT L

=s

··· 0

∞ f T R 0

⎝

st +1 · · · s T −1

0

∞

⎛



s



s

st +1 ···s T −1

st +1 · · · s T −1

= sf (t +1→T ) R (s).

⎝

T −1

⎞

f t ω (st )⎠ dst +1 · · · ds T −1 .

t =t +1

Notice that



⎛

⎛ ⎝

T −1

t =t +1 T −1

t =t +1

⎞ f t L (st ) st

⎠ dst +1 · · · ds T −1

⎞

f t R (st )⎠ dst +1 · · · ds T −1

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Thus, I can treat the information conveyed by the future signals (st )tT =t +1 as one single signal with c.d.f. F (t +1→ T )ω and p.d.f. f (t +1→ T )ω defined above. Now consider the case where st > 1, if the DM processes the signal, he updates his belief and will choose action r if and u p u p only if st +1 · · · s T < u Rp Rs ; otherwise, he chooses action r if and only if st +1 · · · s T < uR p R . Thus, L

L t

L

tst ,Mt (l | L ) = F (t +1→T ) L ( tst ,Mt (r | R ) = F (t +1→T ) R (

uR pR

) − F (t +1→T ) L (

uL pL uR pR

u L p L st

uR pR u L p L st uR pR

) − F (t +1→T ) R (

uL pL

L

)>0 )<0

By Lemma 1, the DM processes st > 1 as

uR pR

u pR L L st u p F (t +1→ T ) L ( u Rp Rs L L t

F (t +1→ T ) R ( u Rp

×

) − F (t +1→T ) R ( uuRL pp LR )

) − F (t +1→T ) L ( uuRL pp LR )  uR pR uR pR uR pR × S T R st +1 · · · s T ∈ [ , ] = uL pL

<

uL pL uR pR uL pL



u L p L st

×

u L p L st uR pR

uL pL

= st .

Similarly, the DM processes st < 1 as,

uL pL uR pR

= <

uL pL uR pR uL pL uR pR

×

tst ,Mt (l | L ) tst ,Mt (l | R ) u p L

L

u p

F (t +1→ T ) R ( uR p R ) − L

×

u pR ) L L st uR pR F (t +1→ T ) R ( u p s ) L L t

F (t +1→ T ) L ( uR p R ) − F (t +1→ T ) L ( u Rp

×

uR pR

L

−1

u L p L st

= st .

In the case where T¯ < T , Proposition 2 implies that in the continuation game where T¯ = 1, the DM ignores signals in [1 −  , 1 +  ] with small enough  which proves the “only if” statement. Using Corollary 1, the result of the non-degenerate g follows. 2 A.3. Proof of Proposition 2 Proof. By Lemma 1, the decision maker processes s1 ≥

 s1 ≥

uR pR

F 2R

uL pL

 F 2L

uR pR



uR pR uL pL

if and only if





 u p  uR pR uL pL R R = > 1. × STR s2 ∈ 0, uR pR uL pL uL pL

(A.1)

uL pL

The last inequality follows from the fact that the strength of the set of signals s2 ∈ (0, s2 =

uR pR uL pL

uR pR uL pL

) is higher than the strength of

, i.e.,



F 2R

 F 2L

uR pR



uL pL > uR pR uL pL

 f 2R

 f 2L

uR pR



uL pL uL pL = . uR pR uR pR uL pL u pR L pL

The last inequality of equation (A.1) also implies that the DM ignores all weak belief-challenging signals s1 ∈ [ uR On the other hand, the DM processes s1 <

 1 s− 1 >

uL pL uR pR

1 − F 2L

 1 − F 2R

uR pR



uR pR uL pL





u p  uL pL uL pL uL pL R R = × STR s2 ∈ ,∞ > . uR pR uR pR uL pL uR pR uL pL

The last inequality is implied by

, 1].

if and only if

(A.2)

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B.T.K. Leung / Games and Economic Behavior 120 (2020) 345–369

 F 2R

 1 − F 2L

 STR s2 ∈

uR pR uL pL uR pR



 > F 2L

uR pR uL pL



uL pL



> 1 − F 2R

,∞

uL pL

;

uR pR

 ;

uL pL



u p R R



> 1. 2

Combining inequalities (A.1) and (A.2) proves the results. A.4. Proof of Proposition 3

Proof. Point 1 of the proposition is directly implied by the Proposition 2. For point 2, by Proposition 2, − > + if and only if:

 u 2L u 2R

p 2L p 2R

1 − F 2L

× 1 − F 2R

 u p  R R uL pL

 u p  R R uL pL

F 2L

u p  R R

F 2R

uL pL

 u p  > 1. R R uL pL

First, note that

 F 2L



lim

u L p L /u R p R →∞

F 2R

uR pR



 f 2L

uL pL = lim uR pR u L p L /u R p R →∞

 f 2R

uL pL

=

uR pR



uL pL  uR pR uL pL

uR pR

lim

u L p L /u R p R →∞

uL pL

which implies

 u L p L F 2L



lim

u L p L /u R p R →∞

 lim

u L p L /u R p R →∞

u 2L p 2L u 2R

p 2R

1 − F 2L

× 1 − F 2R

uR pR

u R p R F 2R

 u p  R R uL pL

 u p  R R uL pL

The result follows given the continuity of − and

F 2L



uL pL  = 1 > 0; uR pR uL pL

u p  R R

F 2R

uL pL

 u p  = +∞ > 0. R R uL pL

+ in p L .

2

A.5. Proof of Corollary 2 Proof. I prove the corollary at the limit where p L → 1. Follow from the proof of Proposition 3,

lim

 + = 1;

lim

− = +∞.

u L p L /u R p R →∞ u L p L /u R p R →∞

It is then obvious that as p L → 1, for all s > 1,

U P (s, M1 ) > U I (s, M1 ) U P (s−1 , M1 ) < U I (s−1 , M1 ). The result then follows from Corollary 1.

2

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365

A.6. Proof of Corollary 3 Proof. I prove the corollary at the limit where p L → 1. Follow from the proof of Proposition 3,

lim

 + = 1;

lim

− = +∞.

u L p L /u R p R →∞ u L p L /u R p R →∞

Now suppose + → 1 and − → +∞. The probability that the DM chooses action r converges to



p L F 1L (1) F 2L

uR pR



uL pL



+ p R F 1R (1) F 2R

uR pR



(A.3)

.

uL pL

On the other hand, if the DM can process both signals, the probability that he chooses action r equals



∞ pL

F 2L

uR pR

 f 1L (s)ds + p R

uL pL s

0



∞ F 2R 0

uR pR



uL pL s

Equation (A.3) is strictly smaller than equation (A.4) as for



∞ F 2ω 0

uR pR

 f 1ω (s)ds >

uL pL s



1 F 2ω

 F 2ω

> 0

= F 2ω



uR pR uL pL

uR pR uL pL

(A.4)

ω = L and R,



uR pR uL pL s

0

1

f 1R (s)ds.

f 1ω (s)ds

 f 1ω (s)ds

 F 1ω (1).

The result follows by the continuity of the probability of choosing action r in + and − .

2

A.7. Reverse wishful thinking To illustrate reverse wishful thinking, I normalize the utility function as follows:

u (r | L ) = −u L < 0; u (r | R ) = u R > 0; u (l | R ) = u (l | L ) = 0. State L is the undesirable outcome which yields weakly negative utility while state R yields weakly positive utility. By analogy to the definition of wishful thinking, the processing strategy of the DM exhibits reverse wishful thinking when + < + . That is, he processes a larger set of information that supports the undesirable state, compare to that supports the desirable state. The following result hold as the Corollary 9 (Reverse wishful thinking). When state L is very undesirable, the equilibrium processing strategy of the DM exhibits reverse wishful thinking, i.e., − > + when | − u L /u R | is big enough. A.8. Proof of Proposition 4 Proof. By Proposition 2,

+ A =

uR pR uL pL +

= B

On the other hand,



F 2R



× F 2L

uR pR



uL pL  uR pR uL pL

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B.T.K. Leung / Games and Economic Behavior 120 (2020) 345–369

 u p  R R



− A =

1 − λA + λA

uL pL uR pR

1 − F 2L

1 − λA + λA

1 − F 2R

 >

1 − λB + λB

uL pL uR pR

uL pL



×

1 − F 2L

1 − λB + λB

uL pL

 u p  R R



×

 u p  R R

1 − F 2R

(A.5)

uL pL  u p  R R uL pL

= − B



where the second inequality of equation (A.5) follows from the fact that 1 − F 2L

uR pR



 > 1 − F 2R

uL pL

uR pR



uL pL

2

.

A.9. Proof of Proposition 5 Proof. First, by Proposition 2,

   A uR pR + δ 1 − F 2L uL pL uL pL    u p   u p  R R R R B A + δ 1 − F 2R 1 − λ − δ + λ 1 − F 2R uL pL uL pL    B uR pR 1 − λ + λ 1 − F 2L uL pL   u p  R R B 1 − λ + λ 1 − F 2R 

− A =

>

B 1 − λ − δ + λ 1 − F 2L

uL pL uR pR

uL pL uR pR

 u p  R R

(A.6)

uL pL

−

= B



A where the second inequality of equation (A.6) follows from the fact that 1 − F 2L

+

+

Now I compare  A and  B . By Proposition 2,



+ A =

+ B =

uR pR uL pL

B λ F 2L

uR pR uL pL

+

B λ F 2R

B λ F 2R B λ F 2L

  

uR pR



uL pL  uR pR uL pL uR pR





A + δ F 2R A + δ F 2L



A F 2L



uL pL



 A > 1 − F 2R

uR pR uL pL

 .



uL pL ; uR pR uL pL

uL pL . uR pR uL pL

+

Therefore,  A ≤  B if and only if A F 2R



uR pR

uR pR

uR pR



uL pL ≤ uR pR uL pL



B F 2R B F 2L



uR pR



uL pL . 2 uR pR uL pL

A.10. Proof of Corollary 5 Proof. Without loss of generality, I analyze only the processing strategy of Alice. First by Corollary 2, Alice processes signal q because

q>

1− p p

× q,

while she processes signal q−1 if and only if

B.T.K. Leung / Games and Economic Behavior 120 (2020) 345–369

q>

p 1− p

×

1 − λ + λq(1 + q)−1 1 − λ + λ(1 + q)−1 p

Define F ( p , q, λ) = q −

1− p

×

=

p 1− p

1−λ+q 1 + q(1 − λ)

×

1−λ+q 1 + q(1 − λ)

367

.

. Its first derivatives w.r.t. p and λ are

∂F 1 1−λ+q =− < 0; 2 ∂p (1 − p ) 1 + q(1 − λ) q2 − 1 p ∂F < 0. =− ∂λ 1 − p (1 + q(1 − λ))2 Moreover,

F (0, q, λ) = q > 0 >

lim

λ(1 − q)

F ( p , q, λ) = q

;

1 + q(1 − λ) p p F ( p , q , 0) = q − > 0 > F ( p , q, 1) = q(1 − ). 1− p 1− p ( p /1− p )→q

Therefore, there exists p − and λ− such that F > 0 if and only if p < p − and if and only if λ < λ− . It remains to prove point 2 of the corollary. First, F is strictly convex in q

∂ 2F p 2λ(1 − λ)(2 − λ) = > 0, 1 − p (1 + q(1 − λ))3 ∂ q2 and the two roots of F (q) = 0 are

−(1 −

p 1− p

)±

 (1 −

p 1− p

)2 + 4(1 − λ)2

p 1− p

2(1 − λ)

(A.7)

.

One of the two roots is negative, which contradicts the fact that q > 1. Hence, there exists a q− , which is the positive root defined in equation (A.7), such that F > 0 if and only if q > q− . 2 A.11. Proof of Proposition 6 Proof. When λ < λ− , Alice and Bob processes both belief-confirming and belief-challenging information. They take the same action if and only if at least one of s1 and s2 is informative.

P consensus = 1 − (1 − λ)2 = λ(2 − λ).

(A.8)

On the other hand, when λ ≥ λ− , they ignore belief-challenging signals in period 1. Therefore, Alice and Bob take the same action if and only if the signals in both periods support the same state, or if s1 is pure noise while s2 is not.

P consensus = λ2 (q2 (1 + q)−2 + (1 + q)−2 ) + λ(1 − λ)

(A.9)

Similarly, the probability that Alice/Bob takes her/his a priori optimal action follows:

 a P default

=

1 − λ(1 + q)−1 − λ(1 − λ)(1 + q)−1 1−λ

 b P default

=

2

(1 + q)−2

− λ(1 − λ)(1 + q)−1

1 − λq(1 + q)−1 − λ(1 − λ)q(1 + q)−1

if λ < λ− if λ ≥ λ−

1 − λ2 q2 (1 + q)−2 − λ(1 − λ)q(1 + q)−1

if λ < λ−

(A.10)

if λ ≥ λ−

First, I prove the first part of the proposition. First, notice that equation (A.8) is increasing and convex in λ. On the other hand, the first and second derivative of equation (A.9) w.r.t. λ are given by:

∂ P consensus = 1 − 2λ(1 − (q2 (1 + q)−2 + (1 + q)2 )) > 0 ∂λ 2 ∂ P consensus = −2(1 − (q2 (1 + q)−2 + (1 + q)2 )) < 0 ∂ 2λ

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respectively. The two inequalities are implied by the fact that q > 1 and λ ∈ [0, 1].35 Secondly, equation (A.9) is clearly ˜. smaller than equation (A.8) for any given λ, which implies that there is a downward jump at λ = λ j Now I move on to the proof for P default , j = a, b. First, part 1 is implied by the fact that both functions a P default = 1 − λ(1 + q)−1 − λ(1 − λ)(1 + q)−1 b P default = 1 − λq(1 + q)−1 − λ(1 − λ)q(1 + q)−1

decrease in λ as λ(2 − λ) increases in λ for λ ∈ [0, 1]. On the other hand, from equation (A.10), it is obvious that there is a ˜. 2 upward jump at λ = λ A.12. Proof of Corollary 6 Proof. Without loss of generality, I analyze the case where s1 = q and s2 = 1/q. When λ ≥ λ− , q ≤ q− or p ≥ p − , Alice and Bob process belief-confirming information but ignore belief-challenging information in period 1. Therefore, Alice processes s1 while Bob ignores s1 and processes s2 . Their beliefs in period 3 are:

 PraL (M3a )

=

PraL (q)

= 1+



PrbL (M3b ) = PrbL (q−1 ) = 1 +

1− p pq p

(1 − p )q−1

 −1  −1

> paL ; < pbL .

On the other hand, the probability that s1 > 1 > s2 or s2 > 1 > s1 equals

2λ2 q(1 + q)−2 , which increases in λ and decreases in q.

2

Appendix B. Supplementary material Supplementary material related to this article can be found online at https://doi.org/10.1016/j.geb.2020.01.005. References Abramowitz, Alan, Saunders, Kyle, 2008. Is polarization a myth? J. Polit. 70 (02), 542–555. Acemoglu, Daron, Chernozhukov, Victor, Yildiz, Muhamet, 2007. Learning and Disagreement in an Uncertain World. Working Paper. Aghion, Philippe, Tirole, Jean, 1997. Formal and real authority in organizations. J. Polit. Econ. 105 (1), 1–29. Akerlof, George, Dickens, William, 1982. The economic consequences of cognitive dissonance. Am. Econ. Rev. 72 (3), 307–319. Baldassarri, Delia, Gelman, Andrew, 2008. Partisans without constraint: political polarization and trends in American public opinion. Am. J. Sociol. 114 (2), 408–446. Baliga, Sandeep, Ely, Jeffrey C., 2011. Mnemonomics: the sunk cost fallacy as a memory kludge. Am. Econ. J. Microecon. 3 (4), 35–67. Bartels, Larry M., 1998. Electoral continuity and change, 1868–1996. Elect. Stud. 17 (3), 301–326. Bartels, Larry M., 2000. Partisanship and voting behavior, 1952–1996. Am. J. Polit. Sci., 35–50. Bénabou, Roland, Tirole, Jean, 2002. Self-confidence and personal motivation. Q. J. Econ., 871–915. Benjamin, Aaron, Bjork, Robert, 2000. On the relationship between recognition speed and accuracy for words rehearsed via rote versus elaborative rehearsal. J. Exp. Psychol. Learn. Mem. Cogn. 26 (3), 638–648. Brewer, James B., Zhao, Zuo, Desmond, John E., Glover, Gary H., Gabrieli, John D.E., 1998. Making memories: brain activity that predicts how well visual experience will be remembered. Science 281 (5380), 1185–1187. Brunnermeier, Markus, Parker, Jonathan, 2005. Optimal expectations. Am. Econ. Rev. 95 (4), 1092–1118. Caplin, Andrew, Leahy, John V., 2019. Wishful Thinking. Technical report. National Bureau of Economic Research. Carrillo, Juan D., Mariotti, Thomas, 2000. Strategic ignorance as a self-disciplining device. Rev. Econ. Stud. 67 (3), 529–544. Coltheart, Max, Langdon, Robyn, McKay, Ryan, 2011. Delusional belief. Annu. Rev. Psychol. 62, 271–298. Compte, Olivier, Postlewaite, Andrew, 2012. Belief Formation. Working Paper. Connors, Michael H., Halligan, Peter W., 2015. A cognitive account of belief: a tentative road map. Front. Psychol. 5 (1588), 1–14. Craik, Fergus, Govoni, Richard, Naveh-Benjamin, Moshe, Anderson, Nicole D., 1996. The effects of divided attention on encoding and retrieval processes in human memory. J. Exp. Psychol. Gen. 125 (2), 159–180. Craik, Fergus, Lockhart, Robert S., 1972. Levels of processing: a framework for memory research. J. Verbal Learn. Verbal Behav. 11 (6), 671–684. Craik, Fergus, Tulving, Endel, 1975. Depth of processing and the retention of words in episodic memory. J. Exp. Psychol. Gen. 104 (3), 268–294. Crémer, Jacques, 1995. Arm’s length relationships. Q. J. Econ. 110 (2), 275–295. Dunning, David, Balcetis, Emily, 2013. Wishful seeing: how preferences shape visual perception. Curr. Dir. Psychol. Sci. 22 (1), 33–37. Eil, David, Rao, Justin, 2011. The good news-bad news effect: asymmetric processing of objective information about yourself. Am. Econ. J. Microecon. 3 (2), 114–138. Enke, Benjamin, Zimmermann, Florian, 2017. Correlation neglect in belief formation. Rev. Econ. Stud. 86 (1), 313–332.

35 First, [q2 (1 + q)−2 + (1 + q)−2 ] ∈ (1/2, 1) as it is increasing in q. Therefore, 1 − (q2 (1 + q)−2 + (1 + q)−2 ) ∈ (0, 1/2). Combined with the fact that λ ∈ [0, 1], it implies the first inequality.

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Evans, John H., 2003. Have americans’ attitudes become more polarized? An update. Soc. Sci. Q. 84 (1), 71–90. Eyster, Erik, Rabin, Matthew, 2005. Cursed equilibrium. Econometrica 73 (5), 1623–1672. Flaxman, Seth, Goel, Sharad, Rao, Justin, 2016. Filter bubbles, echo chambers, and online news consumption. Public Opin. Q., 298–320. Gentzkow, Matthew, Shapiro, Jesse M., 2011. Ideological segregation online and offline. Q. J. Econ. 126 (4), 1799–1839. Goette, Lorenz, Han, Hua-Jing, Leung, Benson Tsz Kin, 2019. Information Overload and Confirmation Bias. Working Paper. Goldstein, Daniel, McAfee, Preston, Suri, Siddharth, 2011. The effects of exposure time on memory of display advertisements. In: Proceedings of the 12th ACM Conference on Electronic Commerce. ACM, pp. 49–58. Golman, Russell, Hagmann, David, Loewenstein, George, 2017. Information avoidance. J. Econ. Lit. 55 (1), 96–135. Heller, Yuval, Winter, Eyal, 2019. Biased-belief equilibrium. Am. Econ. J. Microecon. https://doi.org/10.2139/ssrn.3281674. Forthcoming. Hetherington, Marc, 2009. Review article: putting polarization in perspective. Br. J. Polit. Sci. 39 (02), 413–448. Itti, Laurent, Baldi, Pierre, 2006. Bayesian surprise attracts human attention. Adv. Neural Inf. Process. Syst. 18, 547–554. Jehiel, Philippe, 2005. Analogy-based expectation equilibrium. J. Econ. Theory 123 (2), 81–104. Jehiel, Philippe, Steiner, Jakub, 2018. Selective Sampling with Information-Storage Constraints. Working Paper. Kahan, Dan M., Peters, Ellen, Wittlin, Maggie, Slovic, Paul, Ouellette, Lisa Larrimore, Braman, Donald, Mandel, Gregory, 2012. The polarizing impact of science literacy and numeracy on perceived climate change risks. Nat. Clim. Change 2 (10), 732–735. Karlsson, Niklas, Loewenstein, George, Seppi, Duane, 2009. The ostrich effect: selective attention to information. J. Risk Uncertain. 38 (2), 95–115. ˝ Koszegi, Botond, 2003. Health anxiety and patient behavior. J. Health Econ. 22 (6), 1073–1084. Krizan, Zlatan, Windschitl, Paul D., 2007. The influence of outcome desirability on optimism. Psychol. Bull. 133 (1), 95–121. Langdon, Robyn, Coltheart, Max, 2000. The cognitive neuropsychology of delusions. Mind Lang. 15 (1), 184–218. Matejka, Filip, McKay, Alisdair, 2014. Rational inattention to discrete choices: a new foundation for the multinomial logit model. Am. Econ. Rev. 105 (1), 272–298. Piccione, Michele, Rubinstein, Ariel, 1997. On the interpretation of decision problems with imperfect recall. Games Econ. Behav. 20 (1), 3–24. Prior, Markus, 2013. Media and political polarization. Annu. Rev. Pol. Sci. 16, 101–127. Shallice, T., Fletcher, P., Frith, C.D., Grasby, P., Frackowiak, R.S.J., Dolan, R.J., 1994. Brain regions associated with acquisition and retrieval of verbal episodic memory. Nature 368 (6472), 633–635. Sicherman, Nachum, Loewenstein, George, Seppi, Duane J., Utkus, Stephen P., 2016. Editor’s choice financial attention. Rev. Financ. Stud. 29 (4), 863–897. Sims, Christopher A., 2003. Implications of rational inattention. J. Monet. Econ. 50 (3), 665–690. Tversky, Amos, Kahneman, Daniel, 1973. Availability: a heuristic for judging frequency and probability. Cogn. Psychol. 5 (2), 207–232. Uncapher, Melina, Rugg, Michael, 2005. Encoding and the durability of episodic memory: a functional magnetic resonance imaging study. J. Neurosci. 25 (31), 7260–7267. Wagner, Anthony, Schacter, Daniel, Rotte, Michael, Koutstaal, Wilma, Maril, Anat, Dale, Anders, Rosen, Bruce, Buckner, Randy, 1998. Building memories: remembering and forgetting of verbal experiences as predicted by brain activity. Science 281 (5380), 1188–1191. Wilson, Andrea, 2014. Bounded memory and biases in information processing. Econometrica 82 (6), 2257–2294.